l>? ^^^ i^ : ve5> :35:>s>:::>>3 > i> .^^^ )>^,^ :>0 .a . ":>3»5>"3»i^^ !»!>■■, ^ y^^>Dm^:>^ X> ::Sg) 3)^^ JL^'-^^^^^J' :^3 i't>>: 1^:3:3 >:> ^!»^^::> ;^« / L' THE PROBABILITY OP HITTING AN OBJECT OF AM FORM. «¥■ P. BREGER, Ca^UcLme WArtillerie de la Marine. TRANSILA-TED BT C. A. STONE, Lieutenant United States Navy. TO WHICH IS PREFIXED A DEDUCTIO:^^ OF THE PEOBABILITY CUEVE AND DETERMINATION OF THE PROBABLE RECTANGLE. WASHIl^GTON: aOVERNMENT PBITsTTING OFFICE. 1883. 5782 THE PROBABILITY OF HITTING AN OBJECT OF ANY FORM, The following is a translation of a memoir by P. Breger, Capitaine d'Artillerie de la Marine, published in the Mernoires Militaires et Sci- entifiqnes, 1877, to which is prefixed a deduction of the i)robability curve and the method of finding the probable rectangle. The deduc- tion of the probability curve is taken x^i^incipally from Merriman's Method of Least Squares. Without this previous knowledge the me- moir of M. P. Breger would be hardly appreciated or understood. If a gun be fired over a range a number of times under as nearly as may be the same circumstances of elevation, lateral train, charge of powder, and projectile, a certain difference will always be found in the results of the different rounds, both in range and in dcA^ation. As all these rounds were fired with the same degree of care, the results have equal weight, and as they are discordant, we can never be sure that we have found the absolutely true value of the quantity we seek 5 we must therefore be content with the most probable value, and determine within what limits it is probably correct. PROBABILITY. 1. We give, therefore, by way of introduction, the definition and some of the first principles of probability. The word prohahility as used in mathematical reasoning means a num- ber less than unity, which is the ratio of the number of ways in which an event may happen or fail to the total number of possible ways. Thus if an event may happen in a ways and fail in h ways, and each of these ways is equally likely to occur, the probability of its happening is — — y' and the probability of its failing is -^ry* Thus probability is always expressed as an abstract fraction, and is a numerical measure of the degree of confidence which we have in the happening or failing of an event. If the fraction is 0, it denotes impossibility, if J it denotes that the chances are equal for and against its happening^ and if it is 1 the event is certain to happen. 2. Hence unity is the mathematical symbol for certainty. And since an event must either happen or not happen, the sum of the probabilities of happening and failing is unity. Thus if P be the probability that an event will happen, 1— P is the probability of its failing. 3 3. If an event may happen in a ways and also in a' ways and fail in h ways, the probability of its happening is, by Article 1, — j — rri ) ^^^ since this is the sum of the probability of happening in a ways, and that of happening in a' ways, it follows that if an event may happen in different independent ways the probability of its happening is the sura of the separate probabilities. 4. Let us now ask the probability of the concurrence of two iu de- pendent events. Let the first be able to happen in ai ways and fail in &i ways, and the second happen in «2 a-nd fail in ^2 ways. Then there are for the first event ai+&i possible cases, and for the second «2+&2; and each case out of the «i+Z>i cases may be associated with each case out of the <X2-f &2 cases, and hence there are for the two events («!+&]) («2-|-&2) compound cases, each of which is equally likely to occur. In «i «2 of these cases both events happen, in hi h^ both fail, in ai h^ the first happens and the second fails, and in ^2 hi the first fails and the second happens. Thus we have for two independent events : ai a2 Probability that both happen = (^^iji^^y-^^-p^^)- Probability that both fail = (^^^ft^/ (^^.j^)' Probability that the first happens and the second fails = aih (ai+hi) («^+&2)' Probability that the first fails and the second happens = ^2 h {ai-\-hi) (a2+&2)* And the sum of these is unity, since one of the four events is certain to occur. Now, considering each event alone, the probability of the first happening is 77-07, and of the second .^^ , and since tti «2 Cll O2 (ai+&i) (612+ &2) ai-\-h ^ «2+V we have established the important principle that the probability of the concurrence of several independent events is equal to the product of the separate x>robabilities. Thus, if there be four events, and Pi, P2, P3, and P4 be the respective probabilities of happening, the probability that all the events will happen is Pi P2 P3 P4, and the probability that all will fail is (1-Pi) (I-P2) (I-P3) (I-P4). The probability that the first will happen and the other three fail is Pi (1— P2) (1—1*3) (1— P4); and so on. LAW OF THE PROBABILITY OF ERROR. 5. Although it would seem at first sight that accidental errors could hardly be made the subject of mathematical reasoning, yet the very fact of their irregularity brings them under the laws of probability. Moreover we recognize that they must be subject to the following fun- damental laws of arrangement: 1st. Small errors are more frequent than large ones. 2d. Positive and negative errors (that is, quantities greater and less than the true value) are equally probable, and hence in a large number of trials are equally frequent. 3d. Very large errors do not occur, so that there is always a limit Z, siich that all the positive errors are included between and +?, and all the negative ones be- tween and— L These are the three fundamental axioms which, in connection with the principles of probability, are the foundation of the following rea- soning. 6. Therefore, different errors are not equally i^robable, for in a large number of observations a small error occurs more frequently than a large one, and hence has a greater probability, and an error greater than the limit I has a probability of or is impossible. Hence the probability of an error is a function of that error, so that calling x any error and y its i)robability, the law of probability of error is represented by the equation (1) .............2/=/(^), and will be determined if we can find the form off {x). 7. If a number of rounds be fired from a gun under the same circum- stances, and the ranges be measured, and the mean range found, and the difference between each range and the mean be taken, these differ- ences will be called the errors. Suppose we take as the origin a point at a distance from the gun equal to the mean range and lay off the errors in range to the right or left of this point, according as they are positive or negative. The gun is then supposed to be to the left of the origin. If then corresponding to each error as an abscissa we draw an ordinate of a length proportional to the probability of that error, these ordinates and abscissas will be the co-ordinates of points of the probability curve, and its form will be somewhat like the figure from the axioms of Ar- ticle 5. We have called the difference between the ranges and the mean range the errors. The errors are really the differences between the ranges and the true range ; but this latter quantity we do not know. But, when the quantities are of equal weight, the arithmetical mean is supposed to be the most i)robable value, and therefore this is taken instead of the true value. 8. Let the values of the ranges mentioned above be Ml, M2,M3 . . . M„. If z be the true value, the errors (z—Mx)^ (^— Mg) .... (^— M„) have been committed, and their respective probabilities are ^/i =/(2;— Mi), y.,=f[z—M^) .... 2/n=/('2^— M„); that is, if ii is the whole number of errors, and the error" (.2;— M2) occurs 1X2 times, y^z=z-l—f (z — M2); and we suppose enough ranges to be measured to exhibit the several errors in proportion to their respective probabilities. Now, the probability of committing all these errors is the product of their respective probabili- ties j or if P denote this quantity (2) . . . P=2/i2/2 2/3 . . . 2/.=/(^-Mi)/(^-M2) . . . /(^-M„). This probability P depends upon ^, the true value of the range, which is unknown. If we give to z various values we shall have correspond- ing values of P ; and in the imi)ossibility of finding the true value oiz we can only find its most probable value as given by the n ranges, and the most probable value of z is that for which Pisa maximum. To find the value of z which makes P a maximum we take the natural logarithm of each member of (2), giving (3) . . . logP=log/(^— MO+log/^0-M,)-f . . . +log/(^-MJj whence d¥_df(z-M,) df{z-^i;) ^ /(^-M, )_ Whence -^-y^^^^^+y^^^^^-\- - • • +;^^^:::m:)-^* Letf7/(^-M,)=^(^-Mi)/(^-Mi)<Z(^-Mi), and df{z-n^2) = 'P (^^M2)/(^-M2) d (^-Mg), and we have r7P (4) . . -p- = ri)(^_Mi) + ^i^(^-M2) +^(^_MJ=0. Hence the value of z^ which makes (2) a maximum, is that which satisfies the equation (6) . . . ^(^-Mi) + ^i'(^-M2)+ #(^-MJ=0. • And this equation will furnish us with the most probable value of 0, provided that we can determine the form of the function #. Now, it is universall}' accepted as an axiom that with a number of quantities directly determined and of equal weight the arithmetical mean furnishes the most probable result; that is, the most probable value of z in the case under consideration is M1+M2+ M3+ . . . M, til ^— • n (5) 7 This equation may be written ?i^'=Mi+M2+M3+ . . . M,, whence (7.) .... (^_Mi) + (^— M2) + (^-M,)+ . . . +(^_MJ=0. That is to say, tlie arithmetical mean requires that the algebraic sum of the differences or errors shall be zero. Comparing now (6) and (7) we see that the symbol (P means merely the multiplication by a constant, since the value of z must be the same; hence (8.) . . . (?(^_Mi)4-^(^-M2) + , &c., =74^-Mi) + A:(^-M2) + , &c. Inserting in this the value of (P (2;— Mi), &c., from (5) we have (8a.) . . . _l f{'-^^) + ^/(^-M.) _ _ ^^,. = A;(^-Mi) + ^'(^-M2)4-&c. And since this is true for any number of ranges, it must be true for one, or two, or three ; hence the corresponding terms in the two members must be equal. If, then, x be any error, and y its corresponding prob- ability, so that (1.) y=f(x), we have from (8a) (9.) ....... -^/-{^^ =Jcx or ^^ =lcxdx. f(x)dx y Integrating this we obtain (10.) logyJ^'^+h'. Passing from logarithms, we have (11.) ?/=e*^^V =ce*^^'. Now y is to be positive, and is to decrease as x increases, either posi- tively or negatively ; hence the constant Ic is negative. Placing, then, ^-7t=— /t^j we have (12.)' y=ce-^'^" as the equation of the curve represented in the figure. This equation satisfies the conditions imposed at the beginning of our investigation, for y is a> maximum when x=0, it is symmetrical with respect to the axis of Y, since equal positive and negative values of x give equal values for y; and when x becomes very large y is very small. 9. To determine the constant c. Let Xi, X2, ^3 . . . if,, be a series of errors, Xi being the smallest, X2 the next following, and so on, the difference between the successive values being equal. Then, by Art. 3, the probability of committing one of these errors, that is, the probability of committing an error lying be- tween Xi and ^'„, is the sum of the separate probabilities ce-^'^^^^^ ceh'^xl^ &c., or if P' denote this sum. (13) . . . P '=c(^e-"'-;+e-'"1+ . . . +6-"'^:^ 8 which may be written (13) P'=c2^e-^'^', which denotes the sum of the probabilities of all the errors from Xi to a?n, inclusive. P' denotes, then, the probability that an error lies between Xi and a?n. Now, if i denote the small interval between the successive values of x, and if our ranges are numerous enough so that x may be regarded as a continuous variable, i will be equal to dx-, then (14) ^ ~x e = i e ax, from which, by comparison with (13), we have (15) P'=i / e""% which expresses the probability that an error will lie between the limits Xi and Xj^. Now it is certain that the error will lie between— 20 and +00 , and as unity is the symbol of certainty, we have (16) 1 from which the value of the constant c will be known as soon as we find the value of the integral between these limits. e-^-"^-" dx=2 j e-h'^' dx= -^ ; {a) .-. Ji I e-^'^' dx=zC or f e-^^ (i+^^) Mx = Ce-^' -j . f f e-^' (1+^') Mil dx=Q f e-^' dJi. (b) ' 'J oJ Jo Let hx=z in (a) / c-h^a+x-^) "1°° 1 e-'.' (!+.') Mft=-^^;^^- = 2 (1+^) ' Jo .-. by (b) and (c). 9 Hence from (16) .-^ yjJL u — > — ih VtT (18) 1=^^ orc= Inserting this in (12), we have for the equation of the probability curve, or the law of the probability of error, (19) y=^U T-h e-^"^'. Inserting also the value of c in equation (15), we have (20) V'=^ r\-^'^'dx, which expresses the probability that an error will fall between the limits Xi and x^^. Also, since the integral between the limits —x and -^-x is twice the integral from —x to or from to -\-x^ we have as the probability that an error taken at random is between the limits —X and -\-x, or is numerically less than x. Now (19) is the equation of the probability curve, and the area be- tween the curve and the axis of x is Cydx=^ Ce-^'-'dx. 10. Hence if (16) be multiplied by i it will be the total area of the curve, and if (21) be multiplied by i it will be the area between the lim- its —X and -\-x. Hence expressions (16), (20), and (21) are proportional to the areas of the probability curve corresponding to those limits, and if we regard the total area as unity, the partial area between the limits —X and -\-x will be a fraction given by P' in equation (21). Further, since errors are committed in j)roportion to their probabilities, these integrals and their corresponding areas are proportional to the number of errors which we should expect to find between those limits. If, then, we compute values of P' corresponding to successive numerical values of X in equation (20) they will be fractions proportional to the number of errors numerically less than Xj and at the same time express the prob- abilities of committing an error less than x. As, however, the constant h depends upon the precision of the ranges, and hence varies in differ- ent sets of rounds, we write equation (21) under the form (22) P^ = -4 I e-^'^ dhx= " / e-' dt and compute the values of P' corresponding to successive numerical values of hx or t by the usual methods of the integral calculus. These values are usually tabulated. 10 11. Developing e-^^ into a series by Maclaurin's theorem, multiplying by dt and integrating, we get wliich is convenient for small values of t. If it=-j^, we find P'=0. 42839, and for ^= J, we find P'=0. 52050, whence by interpolation we have for P'=J, ^=0. 4769. Hence hx=0. 4769, or calling r this value of x, corresponding to P'=^, we write (24) r='-^, in which Ji is as yet unknown. 12. Probable errors. By the probable error is usually meant the error the probability of committing which is J. Therefore r is the probable error, to know which we must determine 7i, the measure of precision of the result. Having taken the average ofn equally precise ranges MijMajMg, . , . M^, and calling Zq the arithmetical mean, we have ^ _ M,+M,+M3 +M, . If z is the true value, and if Xi, x-zj Xs . , . . x^ are the errors, their values are (25) Xi=z—Mi, ir2=2;— M2, &c. Also, let Vi, t?2? ^3, &c., be the differences resulting from subtracting each range from the arithmetical mean, or (26) -yirrr^^o — Ml, t'2 = ^0 — M2, &C. If Zo were the true value of the quantity, or Zq=z, then the errors, x, would be the same as the differences, v. But as we can never be sure that Zo represents the true value, we can never determine the errors, Xi, X2, X3 . , . . x^. The values, Vi, V2, v^ . . . . v^, which are readily found in any particular case, we call residuals^ they should be carefully dis- tinguished from errors. 13. The probability of the recurrence of any error, x^ is (19) . y=hi--^e-^'^% and the probability of the occurrence of the system of errors, x^, X2, X3 . . . . x^is the product of the separate probabilities. If, then, h is the same for each of the n observations, we have (27) . . . . P=2/i2/2 ijn = n'^i''7:-i^e-h'^^% where 2x^ denotes the sum ^l^+^/ + ^3'+ +OOI 11 Xow, iu this expressiou h is uDknown, and, further, we liave no means of finding" its exact value. But whatever be its value, P must be a maximum in order to give the most probable value of the range z. AVe are therefore led to conclude that for a giveii system of errors the most probable value of h is that which renders P a maximum. Differentiat- ing (27) with reference to Ji, and putting the first differential coefficient equal to zero, we have (ill whence (29) ?i-2F:S'x2=0, from which (30) 2^=^^h}=^ We have thus the value of li in terms of ^j?^, which, however, we have no means of obtaining, since the errors 0*1, Xo^ X3 . . . . x^ depend upon the unknown true value z. If the number of ranges were infinite, 2.1^ would equal ^r-, and, as the latter is determinate, h would be known. In a large number of ranges, therefore, the equation /«,-=—_,-- will always give a close approximation to the value of h. From (27) we see that Ix- must be a minimum when Pis a maximum 5 calling, therefore, 2v- this minimum value of ^./'-, we will have (31) :^.i^ = :^v-' + ^'S in which Jc^ is a constant to be determined ; and then the value of 7t will be correctly given by insertiug in (30) the value of ^x- from (31). As, however, 2x^ cannot be exactly found, we cannot hope to find the exact value of A-, but must be content with determining an approximate one. Xow, theprobability of committing the systemof errors j?i, x-i^Xs .' . x^ is (27) P = c'^ e-^'^^% or inserting for ^x'^ its value from (31) (32) . . . . V = c" €-^- (^ ^■'-{-^•■-) = c" e -^' ^^"- e-'^^^^. But c" e-'^- ^ ^"^ is by analogy the probability of committing the series of errors I'l ^2, &c., and is therefore an abstract number: calling it Ci, equation ^32) becomes (33) V = c^e-T^'^\ Hence, the law of the probability of any value J: is the same as that of an error x, AVe may regard, then, (33) as the equation of a curve of the same form as the probability curve, and we may show in the same manner that c^ = lu--^.. 12 Therefore we may write (33) (34) :P = M7:-h€-h'^% which shows the probability P of a value 1c. Now, in this equation both h and Jc are unknown, and, as we have said before, we can only expect to determine their most probable values. The value of Ic^ although unknown is fixed and definite ; and hence we conclude as before that the most probable value of h will be that which makes P a maximum. Differentiating, then, equation (34) with reference to hj and placing the first differential coefficient equal to zero, we have (35) . . . ~^=i7c-h e-h'^'-2 hk^e-'''^' hi 7:-h=0y ' whence (36) . 1-2/^2^^ = 0, from which Therefore, from (31) (38) . ^^'=^^' + ^j^.^ as the nearest possible approximation. Since from (30) the value of 2x^ is -^, we have ^''^ w='''+w from which we find the most probable value of h in terms of the known sum of the squares of the residuals, or (^0) '-7w- Substituting this value in (24), we have as the probable error of a single range (41) y=0. 4769. /_?^=0. 6745^/-^. 14. The probability of the arithmetical mean is the probability of com- mitting the system of errors Vi, Vz, v^^ . . , v^, or Po=c'^e-'^^s^ and the probability that the true value of z is Zo-]-x/ is the probability of the system of errors Vi+x', V2-\-x'^ &c., or (42) . P/ = C" e-/i2 S(w+a;')2. Since i: {v-\-x'f=I v'^-\-2x' I v-\-n x'^, and 2' ^?=:0, this becomes (43) yj^^n ^-hH^V^+nx'^) , 13 Hence we have (4-4) . . . Po : P^^ : : e - ^' - ^^ : e -^' i^v'^+nx'^) : : 1 : e - "^* ; that is, the probability of the error in the arithmetical mean is to that of the error x^ as 1 is to e - '* ^^ ^^ For a single range whose error is x', we have however ^0 '- yJ '• - ^ - 6-^'^". Comparing this with (44) we see that if h is the measure of precision of a single range, h ^/ n must be the measure of precision of the arithmetical mean Zq. Therefore Iiq denoting the measure of precision of ^o? we have (J:5) liQ=h Vn. Denoting by Tq the probable error of Zq, we have from (24) ho ro=hr= V 0. 4769: therefore 7'o=~-^, that is, the probable error of the arithmetical mean is equal to the probable error of a single range divided by the square root of the number of ranges. Hence m ro=4-=0.6745j_J!'' . Vn V n (n—1) 15. Probable rectangle. If a number of rounds be fired from the same gun under exactly the same circumstances, and the ranges be measured, and the mean range found, then hj (46) the probable error of this mean may be determined; that is, the limits greater or less than the mean within which the i^rob- ability of a shot striking will be equal to the probability of its not striking. In like manner if the deviations to the right and left are measured, the lateral limits may be determined, v in this case repre- senting the difference between any deviation and the arithmetical mean. But a shot that strikes within the limits in the direction of the range may strike without the limits to the right or left. To determine, there- fore, the rectangle on the horizontal plane passing through the gun within which the probability of a shot striking is equal to the proba- bility of its not striking, we must proceed differently. Since the probability of two independent events occurring together is equal to the product of their separate probabilities, we may take any two numbers whose product is one-half and find the length of the side of the rectangle laterally corresponding to one of these numbers as a probability, and the length of the side in the direction of the range cor- responding to the other number as its probability, and the probability of a single shot striking within the rectangle would be one-half. Thus there may be any number of rectangles, described around the mean point of impact as a center, within which the probability of a shot strik- ing would be equal to the probability of its not striking. If we take the probabilities in the two directions as equal, and equal to the square root of one-half, the resulting rectangle is called the 14 probable rectangle. Of course, the probable rectangle will vary in dimensions for different guns and different mean ranges, and its size will determine the accuracy of the gun for the mean range for which it is calculated. The smaller the rectangle, the greater is the accuracy of fire. 16. To determine the lengths of the sides of the probable rectangle, we let P'= VJ=--0.7071 in (22) p, 2 r^"" ,2 2,, 2 •■=M"-"'»-V1:F«- and calculate the value of Jioo or t^ corresponding to this value of P' by (23). For ^^=0.74 we find P'=0.70468 and for M=0.76 we find P'=0.71754. Therefore for P'=0.7071, ]ix=0.7^3S ^•7438 (47) ... . . ^=-^-. From (24) , 0.4769 r ' therefore (^^) x=^'jr^^^r=lMr, and substituting the value of r from (41), (49) ;z?=1.56r0.6745 /^l, (50) ....... x=lM2^l^. V n — 1 In (50) X is the error, positive or negative, measured from the mean point of impact, therefore 2x or 2a will be the length of the side of the rectangle in the direction of the range, when Iv^ means the sum of the squares of the residuals with reference to the range; and 2^1? or 2b will be the length of the lateral side of the rectangle, when Zv^ means the sum of the same quantities with reference to the lateral deviations. Calling these quantities Ivi and Uv2^, we have for the sides of the prob- able rectangle, (51) . 2a=2.wU^ (52) 36=2.104^^^. MfiMOIRE snr la Probabilite d'Atteindre Un But de forme quelconque. Par p. BEfiGEE, Capitai7ie WArtillerie de la Marine, 15 MEMOIR UPON THE PROBABILITY OF HITTING AN OBJECT OF ANY FORM. OBJECT OF THIS STUDY. The object of this study is the determination of the probability of hit- ting an object of any form. The employment of the ordinary methods fiirniwshes ns a solution of this i)roblem only in two particular cases : first, when we consider a rectangle of which the sides are parallel to the directions in which we measure the errors ; second, when we consider a circle having for its center the mean point of impact, which requires that the mean errors should be equal, a condition which is rarely real- ized; otherwise, one is stopped by the difficulties of the calculation. These methods, which are sufficient in the above cases, become inade- quate under the conditions of actual fire, for which the calculus of prob- abilities would be able to furnish the most valuable and positive informa- tion if we could always surmount these difficulties. Several times, and particularly in his Traite de balistique experimentale^ M. Helie has pro- posed, for practical use, replacing the probability curve, ordinarily rep- resented by the formula of Laplace, by a right line; these results are the development of this iug^enious idea; we thus avoid the almost impracticable management of a compli- cated exponent under the double integral sign. Without doubt this substitution causes an appreciable error, but it is extremely small, and has no practical value when we consider that from the nature of these calculations we cannot hope for an absolute precision; we can, moreover, by very simple means reduce almost to nothing this error, and this con- sideration will without doubt be of a nature to remove the scruples we might have against the employment of this empirical method, which, as we will show, affords a solution of a number of i)roblems unapi)roach- able by the usual methods. 5782 2 17 18 II. PRELIMIIvrARY HYPOTHESIS OF THE CAUSES OE THE DISPERSION OF THE SHOTS AROUND THE jMEAN POINT OF IMPACT. We cousider that the causes which produce the dispersion of the pro- jectiles around the mean point of impact are of two kinds; that the one acts all in the vertical direction, tending to raise or lower the projectile, while the other acts horizontally, tending to move it to the right or left: if, for instance, the target is vertical and has its center (Fig. 1) coin- cident with the mean point of impact, a particular point of impact P will be the result of firing a projectile subjected to the action of a ver- y tical deviating cause, which produces the eri-or OA, and to a lateral de- viating cause, which produces the error OB. This is the preliminary hypothesis adopted; it is evident that we could make any number of others: admitting, for example, that the real directions of the errors are other than oy and ox^ or that there will be a greater number of them. It is true that these suppositions will be hardly justified; but we can imagine that there exists only a single cause of error acting indiffer- ently in all directions, or, better still, that the true law of dispersion holds at the same time of this last hypothesis and of the first; and each of these suppositions assigns a different value to the probability of hit- ting the point P ; if we adopt that of the vertical and lateral errors it is not because it appears certain, but, in the first place, the results to which it leads are sufficiently confirmed by experience; and, in the second l)lace, if we put to one side the two suppositions but little justifiable which we have mentioned above, it has the advantage of assigning to the probability of hitting the central part of the target its minimum value, and, since for this portion the suppositions the most oi^posite give results but little different, it errs on the safe side ; it is this which ex- ])lains why it is to-day universally employed; but that only shows posi- tively that we can count upon it for an approximation. 19 ni. SUBSTITUTION OF A EIGHT LINE FOR THE CURVE OF ERRORS IN THE IN^T:STIGATI0N of the probability OF HATING AN ERROR LESS THAN X; 3IE AN ERROR, EXTRE3IE ERROR. If we take a right line A B (Fig. 2), or rather two symmetrical right lines, A B and A B' for the curve of probability of errors measured along the axis x x from the point O, the probability of having an error oc OTx, to the right of the axis o s is equal to s dx ; in this case O B and O B^ measure the extreme errors, thus all the possible errors are com- prised between zero and OB, and zero and OB', the sum of all the s dx, or the area of the triangle ABB', should be equal to unity, the measure of certainty; hence the equation of the right line AB will be, calling m the extreme error o B, m—x but it should be remarked that x can only vary between zero and m ; the probability of having exactly the error o c will be : (1) that of not passing the error o c or of having an error comprised be- tween zero and +j? will be the integTal of this quantity between the limits zero and x, that is the area of the trapezoid O ADO, and, calling this P, we will have X X? P=--: The probability of not passing the error —x will be of the same value, and, if we consider the errors will be taken indifferently to the right or left of the point o, we will have for the probability of not exceeding the error x. (2) P'= 2x of m iw We know that the mean error, which we will call y, is the abscissa of the center of gravity of the area O AB 5 we have then P'^x [m- 7>r Ulx 20 The area A is equal to J. We find easily m and the same for the other side of the axis of s. Experience also shows that, in firing projectiles, the extreme error is a little more than three times the mean error j* but this is not of great importance ; that which is necessary, is that the probability of exceeding this error should be small enough to be neglected, and we cannot have any doubt in this respect. The mean error is ordinarily the quantity which we have given; replacing m by ^y in the value of P', we have the formula given by M. Helie : ^--1^ 3r V but we will continue to use the quantity m, since it is more easily writ- ten, but will substitute y in any formula of interest. * The abscissa of the center of gravity of the area between the axis of x and one branch of the probability curve yz=-^ e , is fonnd thus : I e X dx 1 -1? x^ Jo ~'9hi -1 i -1? ^ ;j^ Vlt e ax . , ^ 2h 'Wit since To eliminate Ti, we have, calling ?/o the value of y when x = 0; V Tt whence If, therefore, the abscissas of the centers of gravity of the triangle and the area 3 inclosed by the curve are the same, we would have l=~-, and consequently the alti- tude of the triangle would be ^-, or a little greater than y^. Of course the value of Z is determined by the observations, and that of the altitude results from it. Since the mean error is equal to the absicssa of the center of gravity of the area between the curve and the axis of x, we have 1 meanerror=^:^, whence 1 ■y/ ji (mean error)' Since lir = .4769, when the probability is one-half, we have : r = .4769 -/^ (mean error) = 0.8453 (mean error) ; a value sometimes used for the probable error. C. A. S. 21 lY. NEW METHOD OF FINDINO THE PROBABILITY OF HITTINQ ANY AREA. The probability of hitting a particular point of impact is easily- ob- tained. The target can be in any position ; to fix the ideas we will sup- pose it horizontal. Through the mean point of impact o (Fig. 1) x)ass two rectangular axes oy and ox) upon the former measure the errors in range, and upon the latter the errors in direction 5 the point of impact P, of which the co-ordinates are x and y, results from the concurrence of two events : the error in direction x and the error in range y 5 the prob- ability pi of the error x is, as we have seen m—x ^ the probability jpa of the error y w^ill be n—y , in these formulas m and n represent the extreme errors in direction and in range ; the probability ]) of having at the same time the error x and the error 2/, or of hitting the point P, will be the ijroduct i)i|>2? and we will have (m—x) (n—y) , , - an equation in which x can vary from zero to m, and y from zero to n ; it is therefore apx)licable to the first right angle of the axes oy and ox, but it is easily seen that, in order to make it applicable to the other angles, it is only necessary to change the signs of ii? and y. In fact, to find the probability P of hitting any portion of the target inclosed by a curve, y=f (^), it is sufficient to integrate the expres- sion (3) between limits depending only upon y=f {x)^ and we will have, in the first right angle : (4) .... . ^=i^^2j dx£{m-x)xn-7j)chj. A similar formula derived by the ordinary methods would only gi^e a theoretical value, the integration indicated would be impossible 5 it is evident, on the contrary, that equation (4) is easily solved in a great number of cases ; the only difficulty would be when the integration with reference to the first variable introduced a function not iutegrable, which would hardly occur within the limits of practice. However, not- withstanding its simplicity, it requires some i)recautions which should not be lost sight of; the manner in which the subject will be presented in the following paragrai^h is of a nature to dispel all hesitation. 22 SURFACE OF PROBABILITY — ITS EQUATION. DETERMINATION OF PROBABILITY EQUIVALENT TO FINDING A VOLUME. AVhen we assign limits m and n to the two kinds of errors considered, the i)oint8 of imi)act cannot be outside of a determinate portion of the target; but when we use the formula of Laplace this is no longer the case, for the probability curve has then the axis of x for an asymptote^ In the actual case this is evidently an imperfection, for the errors can- not increase indefinitely ; with all others that we know, if the curve wiiich represents them admits a limit of errors, the portion of the target which contains all the points of imx^act will be a rectangle of which the center coincides with the mean point of impact, and of which the sides parallel to the directions of the errors will be double the extreme errors in length. Hence all points within this rectangle can be hit, for the co-ordinates of any point being less than the corresponding extreme errors, they can be considered as the result of one of the possible combinations of the two rectangular errors, varying from zero to their extreme value; with- out this rectangle, on the contrary, there can be no point of impact, for it would require one at least of the co-ordinates having a value greater than the extreme error in that directiou ; by analogy, in practice, we will call it the rectangle containing all the shots, or the rectangle of the extreme errors. Around the sides of this figure, and more particularly near the angles, the points of impact will be less numerous than towards the center, con- sequently there would be required a great number of shots to verify the fact that its form is really rectangular. In the hypothesis of the two deviations this deduction is exact; the size of the extreme errors causes some uncertainty ; but this is a point of no importance. Consider an infinite number of shots fired against a target which can be in any position, but which we will consider still horizontal, as in the last paragraph. Imagine each projectile reduced to the infinitely small dimensions dx^ dy, dz, of a rectangular parallelopiped standing upon the point of the target which it touches ; after firing, the i^rojectiles will be arranged in different numbers upon each element dx dy of the target; they may otherwise be considered as being suj^erimposed one upon the other, in such a manner that they will form a volume bounded by a cer- tain surface, of which the equation is z=:f (a?, y). The probability of hitting any point of the target will evidently be equal to the number of projectiles superimposed upon this point, divided by the whole number of projectiles ; but the volume v of the elementary vertical prism formed by the superimposed projectiles which have touched the given i^oint, and the whole volume Y will be in the same ratio as these numbers. The probability which we seek is then J v^, VV\ 23 equal to ^. The height dz of the projectiles does not affect the result, for if it varies the volumes v and Y will vary it is true, but their rates, and consequently the probability, will not change; it will be of advan- tage to make Y=l; then the probability _p of hitting any point will be simply V ; or the volume of the elementary prism z dx dy. We can then write p=zdxdy. We have already found : (3) (m^x){n-y) from which we deduce (5) zj!!!^=^J^^. This is the equation of the required surface, or the surface of prob- aMUfy, a surface resembling, of course, the empirical form of the adopted curve of probability. If now we wish to find the probability of hitting any portion of the target, or, in other words, an object of any form, we must calculate the volume of a vertical cylinder having this portion for its base, and bounded above by the surface of probability, for this cylinder contains e^adently all the projectiles which, out of an infinite number of shots, would hit the object considered. Thus determining a probability is equivalent to calculating a volume; we have to go through the same operations, and they are the same as far as the calculation is concerned, but perhaps they are not the same as far as the facility of being misled is concerned. YI. THE SURFACE OF PROBABILITY IS A HYPERBOLIC PARABOLOID. The surface of probability represented by equation (5) is a rectan- gular hyperbolic x)araboloid. This is readily shown by moving the axes parallel to themselves until the origin comes over the angle of the rect- angle of the extreme errors in the first right angle of the target ; the equation of the surface becomes then: (6) xy=m'^n^z. If z is constant, we have ii?2/= constant for the equation of the inter- section of all horizontal planes with the surface. This is the equation of a rectangular hyperbola. If either x or y is constant, we have the equation of a right line, as the equation of the intersection of the sur- face by a i3lane perpendicular to x and z, or y and z. 24 YII. HYPERBOLIC OVALS OF EQUAL PROBABLLITY. DISCUSSIOTs" OF THE PORM OF THE CURVES OF EQUAL PROBABILITY. Among all tlie sections which we can make of the surface of proba- bility, those made by planes parallel to the target have a certain inter- est ; they are called the carves of equal prohahiliti/. Thus, each of their points, being situated at the same distance from the target, has been formed by the superposition of the same number of projectiles, and consequently each of the horizontal projections of these points has the same probability. The equation of these curves is obtained by substi- tuting for z a constant value A in the equation of the surface, which will then take the form : (7) [m—x) {n—y) = m^n^A. These curves are the arcs of the hyperbolas which have for asymp- totes the sides of the rectangle of the extreme errors. At the limit these curves coincide with their asymptotes, and become the sides of the rectangle of extreme errors, that which is exactly true in the hy- pothesis of the two deviations, which is that of the curve of probability adopted, requires that we admit a maximum error, for all the points of the right line which bounds this rectangle have the same probability of being hit, which is zero, and without this rectangle the jjoints have no probability; when we approach the center the curve of equal probabil- ity takes an elliptical form, from which it departs more and more as it becomes rectangular. If in equation (7) we make x and y successively equal to zero, we will have the lengths of the semi-axes of the oval : x=7n(l—mnA) Sindy=n (1— mwA), , X m whence, - = — y n The axes of these ovals are in the same ratio as the extreme errors, or as the mean errors, which are themselves proportional to the square roots of the mean squares of the errors. General Didion, in speaking of the formula of Laplace, has shown that the curves of equal probability are ellipses of which the axes are proportional to the square root of the mean squares of the errors ; in fact, the curves nearly coincide, although they are different in their properties. But the deductions which we can make with reference to the real form of these curves are very uncertain ; their forms depend, finally, upon the successive inclinations of the tangent, or in other words upon the successive values of the derivative of an empirical func- tion, and it is very certain that two functions can be nearly coincident without at the same time having the same derivative. It does not sig- 25 nify, then, if these derivatives should be coustantly very different ; on the contrary, it is more probable that their mean will be nearly the same, but still it is not less prqbable that at any given instant they assign to the tangents of the functions from which they are derived directions entirely distinct. The formula of Lax^lace, which in an actual case can only be considered as an approximation, is no exception to this rule ; we have here a remarkable proof of this, for, near the limit of the errors, the curve of equal probability should nearly coincide with a rectangle : in this respect, in spite of its less precision, the right line agrees better with the facts. ** YIII. PEOBABILITY OF HTTTLN'O- VTITHEN" A EECTA^'&LE. AXD CTE^~EEALLY TVITHIN A^^Y CI'EVE SY^CIETEICAL TO AXES PAEAELEL TO THOSE OP THE TAEGET.* The probability of hitting within the rectangle or curve will be, as we have akeady shown, proportional to the volume standing on the rectangle, or the area inclosed by the curve, as a base, and bounded above by the surface of probability. It will be then generally of the form (8) P=B,^, in which B is the area of the base, and z the average height ; so that B^ represents the volume stauding on the base B. The value of z for any point of the base is given by (5) ^ (m-a?)(n-y) from which, its average value for the required area can be found. In the case of a rectangle, if c and c\ d and (V represent the co-ordi- nates of the four corners, relative to the extreme errors, we easily deduce, (^c'-cmd'-cV') P=- 4:ni-n- If this rectangle is symmetrical to the axes of x and y, we wiU have for a rectangle of double dimensions, having its center at the mean point of imx3act, m'-}i- ' and calling a and h the half sides of this rectangle, ^ /'2a a-\ r21) 'b-\ *TMs discussibn is considerably abbreviated from the one given in M, Breger's Memoir. C. A. S. 1^ 26 Substituting for m and n^ Zy and Zy'^ we have the formula deduced by M. Helie in his Cours de Balistique : ^^^ V3r vy w vv* These two last expressions can also be obtained by remarking that, in order to strike within a rectangle having its center at the mean point of imi)act and its sides parallel to the axes, it requires the concurrence of two simple events; first, to have an error in direction less than a; second, to have an error in range less than h. The probability required is then the product of the probabilities of these two events, of which the values are known (2) and (2)i. If the sides of the rectangles are proportional to the extreme errors, P will take a more simple form ; calling ^ the ratios — and -, we have (10) P=(2 Ic^^f^ and calling W the ratios - and —j, (10) ^ = {%lc'-iW^)\ The plans of powder magazines, store-houses, public buildings, &c., are generally of rectangular form ; and these expressions can be easily applied to them under the circumstances of elevated fire. The formula P=B^ will give easily the probability of hitting a circle, an elUpse, a rhombus, &c., these figures being situated entii'ely in the same right angle of the target, and having their axes in the required directions. Without enlarging upon these questions which have no need of any de- velopment, we will make, however, one remark which has some impor- tance ; we can, as we have seen, find the volumes intercepted between the rectangular hyperbolic paraboloid and the bases of different forms without the help of analysis by using the remarkable property* of the center ordinate, a property little known if one judges by the expressions for such volumes in the standard works. In using the right line as the curve of errors, we can, perhaps, render the stud}^ of the probability of fire accessible to a greater number of persons, at least in that which is essential. IX. PROBABILITY OF HITTING A CIRCLE OF VTHICH THE CENTER IS THE MEAN POINT OF IMPACT. The probability of hitting a circle of radius r, of which the center is at the mean point of impact, is obtained in finding the volume of the cylinder having this circle for its base and bounded above by the sur- * The remarkable property referred to is that the area of the base nmltiplied by the center ordinate gives the intercepted volume. C. A. S. 27 face of probability 5 we will use for this formula (4). As tlie y must satisfy all points of the curve, the first integration will be between the limits zero and ^y^—x^\ that with reference to x will be between the limits zero and r j this will be the probability with reference to a quadrant of the circle situated in the first right angle of the target j for the whole circle we will have then : the integration is effected by ordinary methods, and we find : (11) 2r' mn (i - 2r 2r 2 3m 3n or, in terms of the mean errors : _2r2/'r_2r_2r (ll)i This expression has the considerable advantage of being applicable to the ordinary case in which the mean errors have different values. In those which have been determined thus far we have been obliged to suppose these two errors equal, on account of the difficulty of integra- tion. When this equality really exists we have, calling K the equal ratios — and —J m n (11)2 -Kl-f+f) These formulas which* we have deduced, and those which we will de- duce, can be given a precision nearly absolute by means of a correction which we will determine hereafter ; but this correction is generally so small that in practice it is entirely unnecessary. X. PROBABILITY OF HITTINa DIFFERENT TRIANGLES. The probability of hitting a right-angled triangle, such as OAB (Fig. 6), will be the volume of the cylinder having this triangle for its y Jia G J I / c /^ L, K o ^ B X 28 base. Calling OB and AB, h and h, we will have for all points of the right line O A, 2/=^^ j the probability which we require will then be P: which easily gives 1 f*^ r~ y) ^ ^ 2mn\ 3?^ Zm 4mnJ' In the case where the triangle considered extends as far as the side ED of the rectangle of extreme errors, as OOD does, we make 6=m, and we have (1^) • • • ^=6-»0-a)- For a triangle, such as OBF, of which the hypotenuse coincides with the diagonal of the rectangle of extreme errors, the ratios — and — will be equal, and calling them K, formula (12) becomes, after reduction, (14) . p^(?K-^! We have already seen that (10) the probability of hitting a rectangle of which the sides are 2h and 2h is Pi=(2K-E:2)'; we see easily from this that the probability of hitting the triangles, such as OFB and OFI, is the same, and that the whole target is divided into eight triangles, such as ODE, OEG, &c., which have the same probability, namely, g* ^ We arrive at the same result by a different method. If we wish to find the probability of hitting the line IF parallel to the axis of x, and limited by the diagonal OE, or, rather, the rectilinear element IF%, we must integrate the expression which gives the probability of hitting a point (3) between the limits zero and ii?i=IF. Substituting the constant value ^i=FB for y we have, thus: dp'=^ ^^- ±^ay ; in the same manner the probability of hitting the element YBdx will be ( m-xi)( nyi-^) df'=^ ^>^ ^^-^dx, 29 we will have, consequently, dy #' («-,,)(m..-f) observing that n—yi~n~yi~''dy^ we see that the denominator of the fraction ^S"i^ equal to its numer- ator, and that dp"=d;p' ) the two triangles O B F and O F I being composed of the same number of elements xdy and ydx^ having two and two the same probability of be- ing hit, it follows that Zd])"=Id^', or, P''=P'. This equality is applicable to all the right-angled triangles described upon the diagonal of the rectangle of the extreme eri'ors, even when their bases do not coincide with the axes of x and y^ and this does not result from the emi)loyment of the right line as the curve of i)robabili- ties. It is always true in the hypothesis of the two rectangular devia- tions, which should be the representative function of this probability. This function, indeed, contains always a constant which must be de- termined by experience ; this will be the square root of the mean of the squares of the errors ii^ the mean error ^, the probable error Y, the ex- treme error 3^/, &c.; in a word, a particular error c occupying a deter- minate position in the series of errors such that the variables x and y X If can be replaced by other variables —and -75 it is then clear that, if in two functions of the same form we can make equal to each other the X 11 ratios -and-,, since they represent equal probabilities 5 it is this which takes place in the case we will consider hereafter. We see, therefore, that the probability of hitting the right triangles, such as OBF, OFI, FAH, becomes known the same as b}' the formula of Laplace; that is, the half of the probability of hitting the rectangle constructed on their hypotenuse as a diagonal. The probability of hitting the triangle OAL having its angle at the origin, the side opposite the angle parallel to oy will be the difference of the probabilities OAB and OLB ; in particular, that of hitting the tri- angle OKE, which extends to the extreme error DE will be, calling h and li' the heights of the triangles OGD and OKD ; (^^^ Q\ n 4.1V' )■ 30 The different formulas contained in this paragraph are repeated in terms of the mean errors ;' and y'; the letter K' representing the ratios h b -7 and— when these ratios are equal: (13)1 OCD P= IsQ-wd' (14)i^ OFB P= g(^ ^ J. _ 1 fh-h' ¥-W^ - 18V r' (15) OCK P: i2r -)• The probability of hitting the triangles occupying with reference to y the same positions which these triangles occupy with reference to the axis of X is obtained by changing htob and y to /, i XI. PROBABILITY OF HITTING A TRAPEZOID OR ANY RiaHT-LINED FIGURE. To find the probability of hitting a trapezoid ABOD (Fig. 7), we call h the abscissa OE of the point of intersection of the right line A B with the axis of x, and h the ordinate OF of the point of intersection of the same line with the axis of 2/ j then the equation of the right line AB will be, if we call c and d the abscissas of the points A and B, the probability required will be 1 r^ r-h+^ 31 and performing the operations indicated: (16) P = m^w 3b (. n—h m Ti 21) y^. l + id-oQnn-^e-M 2 yj This formula is complicated, but from it we can deduce a number of others more simple j thus, in making d = b yve have the probability of hitting a triangle such as ACE j in making e = o. we have the proba- bility of hitting the trapezoid ODBF; multiplying by two and adding to it that of hitting the rectangle ABCD (Fig. 8), we have the proba- 1/ yf C o jt ~ Ji — !b ; bility of hitting the gable-end of a house in direct fire. If we make at the same time d = h, and c = o, we have that of hitting a triangle such as OFE (Fig. 7). In each case the formula is simplified more or less; in the last case, for example, we find for the triangle OFE, (17) P=: 2mn 7i h Jih \ or in terms of y and y' (17), Jib 18// ,(i- "9r h lib ^y' ' 108// '> for the rhombus formed by four symmetrical triangles situated in each right angle, we multiply by four the value given above j in the particular case where h—n and b—m^ that is to say, when the angles of the rhombus are at the middle points of the sides of the rectangle of extreme errors, we have for P the fixed value f j one has five chances out of six of hit- ting it, while the surface is only half that of the rectangle, which con- tains all the shots. The right line FE (Fig. 7) can take any position we wish ; we should only remark that li and b can become negative j thus AB and BGr (Fig. 10) being the sides of the rectangle of extreme errors, we will have the probability of hitting the triangle EFG in mak- 32 ing <^=mand l)=c in the general formula (16), and in observing that h=OD is negative ; that of hitting the triangle HIJ by subtracting from the known probability of the rectangle OKU that of the trapezoid OHIK, . ^/_ ^1 / c ^ and we will obtain this with the same formula in which &=0T is nega- tive, and in which we will make e=o and h=-, — -^ j and in the same manner for the triangles or the trapezoids occupying the most diverse positions, thus there will be no dif&calty in calculating the probability of hitting any right-lined figure. But these remarks are particularly useful when the object extends beyond the rectangle containing all the shots, we then calculate similar probabilities, as, for instance, when we wish to hit the space comprised between the two right lines LM and NP. ^ XII. PROBABILITY OF HITTING AN ELLIPSE— THE SAME QUESTION SOLVED BY THE EMPLOYMENT OP THE ORDINARY FORMULAS IN THAT WHICH RELATES TO THE ELLIPSES OP EQUAL PROBABILITY — PARABOLA. The probability of hitting an ellipse whose equation is y=- y/a^—x^^ of which the center is at the mean point of impact, and in which a and & are the semi- diameters, is obtained by solving the equation 4 /*« Pa which gives (18) . . . ~ mn\2 3m 3n 4:mny'' or in employing the mean errors, 2ah/^7z 2a (18). P= 3cih\ 3^n 33 We see making a=:h we have the expression already found (ll)i for the probability of hittin^^ a circle. When the ratios — and - are equaL m n we have, callins: them K. P=-2K^Q-Jk+^K'), this, as we have already seen, is the probability of hitting a circle when iw=w and when K= — Thus the probabilitv of hittin^r an ellipse of which the axes are proportional to the mean errors, is equal to that of hitting a circle of which the radius is in the same ratio with the mean errors supposed equal in the two directions. This property is true whatever curve of probability is adopted, because we can show that in such an ellipse, first, the right lines F D and D E (Fig.ll), parallel to 1 y" J? ~°~~~-^ jyX -^"~\ ^/ \ e J. the axes, and intersecting the diagonal of the rectangle of extreme FT) T)F errors, are such that =^7-5 second, that these right lines, and con- r r sequently the right lines HF and EI. have the same probability of be- ing hit. It results from this that these ellipses are divided by the axes ox and 03/, and the diagonals of the rectangles of the extreme errors into eight sectors which have the same probability of being hit, whatever be the curve of probabilities employed ; we see thus that the probability of hitting within the ellipses of equal probability found by the formula of Laplace becomes known, since we know how to find that for a circle for which the probability is the same. We can find the probabihty of hitting these ellipses in a more direct manner. Placing the lormnlar of Laplace under the form 8=- l^e ^^ in which the constant v^ is the mean of the squares of the errors : the probability of having an error x will be i>i= z2_ «^^2^ dx\ 5782- 34 that of having an error y will be and that of hitting any point, or rather an element dxdy^ will be \ We have the same probability of hitting each of the points of the curve of equal probability j particularly that which forms the intersec- tion of this curve with its major axis, for which x equals the semidiame- ter a and y equals zero, will have the probability of being hit: 1 -^ This value can then be taken for all the points of the arc of the ellipse. The area A of the curve is rrahj and if we call h the constant ratio ^, we will have A =V^, whence, taking a as s> variable, <?A= — ^^ — , which is the area comprised between two ellipses infinitely near, for which the ratio y is constant, we can admit that the probability d^ of hitting this particular elliptical element is equal to #P times as many dA elements dx dy as there are in this surface; that is to say, to ^"^^^I*? replacing d/^V and dA by their values, we find whence we have 1 — ^ dV=—jje 2^*^ ada^ uu'k ' 1 r^.^ uu'Jc J ' P: which, being directly integrated, gives P: -1^.0--). and, since in the ellipses of equal probability we have -=— ^, it becomes, which is the probability of hitting a circle of radius a when the mean squares of the errors are equal, and consequently the mean errors. It is unnecessary to remark in this case that the circle is a curve of equal probability. 35 For the parabola of which the equation is y^=2px, we will have as the probability of hitting the portion situated between the curve the axis of a; and the right line x=b, P=^— -, I {m—x)dx I '^^^{n—y) dy. We easily find 26. ^'' mn\3\p 6m 2n Smn J If we call Ji the ordinate corresponding to x=h: p_ hh /'2 2& h hh\ mn V3 5m 4in QmnJ' When &=m and ?i=7t, P has the fixed value ^) the parabola has then the equation XIII. PROBABILITY OF HITTING THE HYPERBOLIC OVALS OF EQUAL PROB= ABILITY. Requiring the probability of hitting one of the curves of equal prob- ability which we have already determined, and which are formed by four equal and symmetrical arcs of hyperbolas, we know that these hyperbolas have the equation (7) (a) {m—x)[n--y)—mH^A] but referring them to two axes formed by the sides of the rectangle of extreme errors, they take the form XT=m2/i2A. From equation (a), the value of x when 2/==0, is x^m—mhiJ^, Letting c=m-?iA, we have for the probability of hitting within one quadrant of the oval, ■^"^^SF / (^i--^) clxj ^n-^y) dy, or Mf- I ( m—x ) dj)» S6 Integrating ^=2«< 2^-2+'^ '<*«•«> Bubstittiting the raloe of e, we have or for the probability of hittifig aoy where within the whole oval j P2= (1 — mVA^+ m^^A^ log. 7nhi'A'). KoTE.— This demonstration is different from that in M. Brdger^s Memoir, but the result obtained is the same, 0. A. S. If in this relation we make A=0, P becomes equal to unity, that is to say, to certainty ^ in fact, this supposition reduces the curves of equal probabilities to their asymptotes, which form, as we have seen, the sides of the rectangle of extreme errors, which contains all the shots j giving A the value — ? which corresponds to the case where these curves are reduced to a single point, the center of the target, we easily find P=0. The quantity A is seldom given, but we can write it as a function of known quantities; if, for example, the hyperbola is determined by its I)oints of intersection with the axes of the target Oy and Oil?, we will have, by making x 'dud y successively equal to z^ro, A= — ^ — and A^ f * whence we find m n m n If we call K the quantity mnAj the probability of hitting the target within the curves of equal probability will take the simple form P^1«-K2^K2 log. K^ + XIT. I'ROBAIBIILITV OF HITTINO AKY CIRCLE— AFPLICATIOlS* TO EIRE AOAINST MOKCRIEPI' BATTERIES. To determine the probability of hitting a circle of radius r, the co- ordinates of the center of which are a and ft, such that all of the circle is in the same right angle of the target, we write from equation (8) 37 in which B is equal to -r^, X to m—a^ and Y to n—h. Substituting these values, we have (i») • p=^<'»-«) ('»-'')' or in terms of the mean errors ^19)i ^=-8i5p ^^'^-"^ (^''■-^)- If the center of the circle is on the diagonal of the rectangle of ex- treme errors, the ratios — and - will be equal, and, calling them K, we m n ^770 7 have P=— (1-2K+K^). mn In the case where the mean errors are equal, and the circle tangent to the four sides of the square formed by tlie extreme errors and the axis of X and y^ we find for P the fixed value ^^ and for the four sym- metrical circles, ^ . These formulas can also be obtained by the solution of the expression P=^-^2 / ^« / (^>i-/>cosa-a) (7i-/?sina-&)/>^^, ' which we obtain by moving the axes to the center of the circle to be hit, and using polar co-ordinates. This requires very simple calculation, and often has the advantage of being useful when the circle is not all in the same quadrant of the target. Jf the center is on the axis of y we make a=0; the integration with respect to p is between the limits zero and r. Eemarking, then, that the probabilities of hitting the half circles situated on each side of the axis of y are equal, we make the second in- tegration between the limits — ^ and -j-^, when multiplying by two, we find and if the center is on the axis of x, (3;:-4r) { m-^a)r^ Major Moncrieff has proposed replacing casemated batteries by cir- cular pits, lined with masonry, much larger at the bottom than at the top, in which are mounted guns upon carriages of his invention, which 38 recoil below the surface. The pits are connected together by under- ground passages. This system appears to be in great favor in England, where it has been tried. Among the methods proposed by the artiller- ists of this nation, we find the arrangement of the pits in groups of lives. Let us find the probability of hitting one of these groups of fives, considering as good only those shots which penetrate into a pit, sup- posing that we aim at the center of a group. Let us take the diameter of the pits as 6 meters, and suppose that the center of the central pit is the mean point of impact, and the origin. Let the co-ordinates of the centers of the other four pits be ±5 meters, so that they will be arranged symmetrically with reference to the origin. Formula (11) gives immediately the probability of hitting the central pit, and (19) that of hitting any of the others 5 letting r=3, a=b=^, and we have ^^-y/ 3r/' 3fr' 2fr'^ for the central pit, and _ rr __ 5- __ Stt 257r J^a = — , 3-/2 3PP + 9PP for each of the other four pits. Therefore, the probability of hitting the battery, or of having a projectile enter a pit, will be the sum Po-f 4Pa, which we can write, taking 7:= 3. 14, V'-^) "— ^^/ 2 / /2 "I" 9 /2' m7 22.3 22.3 . 35.4 yyl We should call attention to the fact that this formula is only ap- plicable when the five pits are contained within the rectangle of the extreme errors, which requires that 7' and y' should both be greater than 2"^. 66 ; if, for example, the mean lateral error is 3 meters and the error in range 20 meters, we find that P=0.206; we would have 21 chances out of 100 in our favor, and, if we fire a great number of shots one-fifth of them would enter the pits. We find the probability of hitting the central pit is 0.048, and that of hitting each of the others is 0.041. In consequence of the greatness of the errors, these two probabilities differ but little. It results evidently from this, and whenever the mean errors are large it would be better to aim at the center of the battery than at each of the pits in succession, and if a number of pieces are firing we would not assign to each of them a particular object to be aimed at, but would aim them all at the central pit. There is also another result to be deduced from this, which is that it would not require many more shots to silence the whole battery than would be required to silence a single isolated pit, for the most probable case is that the projectiles that hit would be divided amongst the pits propor- tionally to their respective probabilities, which at the same time agrees with good sense and the theory. It is true that the probability of this 39 division being perfectly exact is very small: but it will differ bur little from This, and will rapidly approach it as the nuuiber of shots increase. Therefore, we will have a very great probability that the number of shots which will hit each of the pits will be at least ecjual to a number slightly less than that assigned by the most probable division. If. for example, we tire lUU shots, under the above conditions, it is logical to admit that the central pit will receive live shell and the others four; but there is a very great probability that each pit will receive at least three. XT. PE OB ABILITY OF HITirN-G- A SHIP ^lOTTN-G- OBLIQUELY ^ITH EEGAED TO THE LI^'E OF FERE. In order to find the probability of hitting the deck of a vessel moving in an oblicjue direction with regard to the plane of fire, we should take account of the exact form of the deck : but we coasider that this pre- cision is useless in practice, and that it is sufiicient to consider a rectangle of which the dimensions are a little less. T\'e will take for the length that which is called the lengfJi leticeen pcr]jendlculars. and for the width the beam at the waterdine. information which we can easily procure from maritime ])ublications. Simplified thus, the prob- lem, like must of the preceding, can be solved with difiiculty by the ordinary methods. TTe can only, by other means, divide the rectangle to be hit into a series of small rectangles, of which the sides are parallel to the direction ot the axes, and calculate separately the probability of hitting each of these, an operation impracticable unless we have a great deal of time at our disposal ; even then we would neglect a series of triangles, and. in order to have a good approximation, it would be necessary to carry the subdivision of the rectangles very far. In this way We would lose the benefit of the precision which ought to follow the use of the regular formulas. The ciuestion is. on the other hand, easily solved by the methods of this article : we have, in all eases, to find the probability for trapezoids and txiaiigles. which, after what precedes, offers no difiiculty. It is evident that in consecjuence of the relative size of the ship and the magnitude of the mean errors the figure to be hit can take very different forms, for the precision of the pieces of the new artillery makes the object extend beyond the rectangle of the extreme errors, which con- tains all the shots, and consec[uently gives different forms to the portion of the object to be considered. However, if in real fire we use directly the mean errors given by the tables of fire we may make a great error; these quantities may vary considerably in consequence of errors not inherent in the piece, TTe conclude from this that the object to be hit, if large, is more fi'equently than we wuuld believe contained entirely in the rectangle of the extreme errors : thus. also, when we wish to use 40 elevated fire by diminishing the charges, we will find that the errors in direction are notably larger. However, the mean error in range is al- ways much greater than the mean error in direction, consequently on a horizontal target the rectangle of extreme errors has a very elongated form. It results from this, that in practice three cases present them- selves, which we will examine successively. In the first case the mean error in direction is very small ; the proba- bility of hitting the ship IJ is then reduced to hitting the parallelogram EFGH, and one obtains it by multiplying by two the sum of the proba- bilities of the trapezoids OKGP and OKFL. Continuing to call for the present /t and h the distances OK and OS, the probability of hitting the trapezoid OKGP will be given by the general formula (16) XI, in which h will be negative, d=m and c=o; the same formula will give the probability of LFKO, but in supposing this trapezoid placed in the first quadrant we see that h has the same value, that b becomes positive, while d=m and c=o. Consequently when we take the sum of the two ex- pressions, all the terms which contain b to the first power, having con- trary signs cancel each other, those containing c become zero, and we can directly write for the probability of hitting the ship If now we call 21 the width of the vessel and w the angle which its axis makes with that of x, we have b. sm (o and h: COSio J these values substituted in that of P give (31) P=l( 2» m^tan^ «> W V cos <o cos* iii Ql > 41 and iu terms of the mean errors; (21)i P= _i r ^r' 3/\cos io 3 cos^ rHan^ 21 > When the ship makes an angle of 45° with the line of fire, this formula becomes and if (o equals 60°, ly/2 l(v ly/2- 3/ 27^2 > 3/=^V^ 3 21/ Finally, if w is equal to 0^ or to 90^, the rectangle to be hit will have its sides i)arallel to the directions of the errors, and we can make use of formulas already found ; in this last case it will be useful to know the length of the ship. Most of the ironclads of the squadron of all powers have at the water-line a width but little differing from 17 meters. For precision , we will take the Eichelieu, to which this width applies, except the deci- mals, wUich we will neglect, and of which the length between perpen- diculars is about 98 meters ; if the fire takes place from a distance of 1,041 meters, with the rifled shell of 22 cent., charge 1 kil., conditions under which the mean lateral error from the tables is 1.90™, and that of the range 27"^.9, we find the following results : for a>=0o 450 6O0 90O P=0.17 0.27 0.50 0.84. We see that we would greatly deceive ourselves if, when a ship makes an angle of 45^ with the line of fire, we take for the probability of hit- ting it the mean of the extreme probabilities. The preceding formulas are applicable to all cases in which I is equal to or greater than ^y sin w ; if I is less than 3^ sin oi, the point P (Fig. 16) falls without the ship, as in Fig. 17, at D. We obtain the probability required by subtracting' from that for OABD the j^robability for FOD, and adding to the dif- ference the probability for E A, and multiplying the result by two. 42 For tbis we can use the general formula for a trapezoid, and, calling li and h the lengths AG and OF, we find for the trapezoid OABD, Pi= n — mil mn 3Zr+3&' h 2 m^li\ the triangle FOD, 3m^ ^3m^ nl) ' m~ bVi ^ + mil mn 36 + 3&" 7j "2' m^h\ r26V' consequently, _ /i /wft^ 6/fc 7^& &2/i ^ 2m7i\ i>i-i?2-^^^2|^3— 2-3,^- ^ + 12l?^^+ ^^ 3F / For the probability of the triangle AGE we have h f n¥ hh nl) Ifli \ '3m' 3m"^m"^12mV' and calling P=2 {pi—P2+P3)i the probability of hitting the ship will be ^ ^ 3n\4.m' m^ h hj' This, in terms of the mean errors, of the width 21 of the ship, and of the angle <y, gives ,oox p_ ^ r l_ ? ^r' cos ^ 3r sin c.-\ ^""^^ • 27r'^ cos^ a> V36^2 sin^ «> 3;- sin ^"^ I I J' When a>=45^ we will have _ 4^ /-J^ ^/2 _v_ ir_\ ^'-27/^18^2" 3r '^l^/2~l^/2J' and when it is equal to 60^, 4Z^ ^ 2^^ 4? V 3r 73 \ /' ^^-27^27^* 3rV3^ r ^ / Thus, the fire being against a ship of the same type as the Eichelieu, and generally against an iron^clad of 17 meters width, but with the gun of 16 cent, of model 1864, at the distance of 3469 meters, for which the mean errors in direction and range are 4".9 and 30 meters, we will have, according to the value of w : forw=Oo 450 6O0 900 P=0.18 0.24 0.32 0.69. We can still use these formulas when 3^ becomes greater than L cos «y+7 sin w^ although, strictly, we would only have the right to use them 43 up to 3;'=L cos (o — I sin w, but the error which can result from this extension is so small in practice that it will not be of consequence. Beyond this, the rectangle replacing the deck of the ship will be entirely comprised in that of the extreme errors. We are then led to a formula less simple than the preceding ones, whose simplicity is very satisfac- tory, considering the nature of the question. We can still employ the division into trapezoids and triangles, but we arrive more easily at a result by taking as the axes of x and y that of the ship and a perpen- dicular to it. For greater simplicity we will consider the axis of the ship at an angle of 45^ with the line of fire; this will give between the former co-ordinates and the new ones the relations : x=(^-Y) Vf and 2/=(X-f Y) Vj: JE^.18. Substituting these values in the equation of the surface of probability, and finding then the probability of hitting the rectangles FDBE and AIHG, we have Pi=^?^. j^^xJlm-(X-T) ^/2] [m-(X+Y) V2]dT, whence, to which we must add four times the probability of DOF, giving ^ ' ^ mn L 3 \m n J Qmnjy 44 a comi)licated result, but in applying it to a single type of vessels it is reduced to a formula of four terms, with numerical coefficients. This reduction we will not make here; we will only consider the general methods which can be simplified in the works on the study of any i)ar- ticular kind of fire. We will give another solution of this same ques- tion; it is less exact as a demonstration, but it gives practically the same value; moreover, it is much more simple. Generally iron-clads haA^e a length greater than five times their breadth. In considering them as formed of five squares we neglect only the extremities, the probability of hitting Avhich is very small. These squares, except the one which has its center at the origin, are in the conditions indicated in YIII ; that is to say, they are arranged in groups of four elements symmetrical to two axes through their center and parallel to those of the target. We can then apply the relation P=BZ, XYB or, what is the same thing, P=— ^-^, the co-ordinates of the points M and N relative to the extreme errors are Xi=m— Z-v/2; X2=m— 2i-/2, Yl=7^-^^/2'j Y2=n-2ZV2, and for each of the squares B=4?2, we find thus, for the probability ot hitting the four squares M, N, P, and Q : 472 _ Pi=,,^ [4:mn-6lV2{m-{-n)-{-20l'], We know the probability Pg of hitting the square (a), and we finally find ^ 41' r 19lV2^ 12W1 (23) ^=m%i^'^''— 3~(''^+''^+T"J' or in terms of the mean errors, 472 p _ 121 Z^ 1 (23), ....... P=gjpp[45;./_19?V2(x+/)+--g- J. Nothing prevents the employment of the same method for ships in which L = 6Z ; it is sufficient to add to P the probability of hitting two squares s, and two squares r of which the sides are Z, we obtain a for- mula similar to the above ; the numerical coefficients alone have differ- ent values, and it is evident that it would also be similar if we substitute for the ship a rectangle such that j is any whole number, which will always be possible by neglecting small portions at the extremities for which the probability is insignificant. The formula (23)i applied to fire at the Eichelieu, with the piece of 16 cent, from a distance of 6,848 meters, for which the mean errors in 45 direction and range are l-j-'^.T and 42 meters, gives for the probability of MttLDg this ship when it is at an angle of 45-, O.IG; we easily find the values of P for a; = () and cv==90-, and we have : cu=^(P, 45-, 90^, P=:0.13. 0.16. 0.25. In comparing tliese results with those obtained at distances of 3469 and 1041 meters, we see that the probability increases more rapidly when w approaches 90^ : that is. when the ship is perpendicular to the line of fire we have the least chance of hittiog its deck. A.t 45" the chances are a little augmented: it is beyond this that the increase be- comes notable : at 60" the probability is a little below the mean of the extreme probabilities, and at 90-- it has its maximum value. TVhen the mean error in direction is very small, the superiority of fire at 90- is enormous : i3Ut this superiority, although always sensible, diminishes as the mean error increases, or, for the same piece, diminishes with the distance. If, for example, the fire is from a battery^, we see the consid- erable advantage that there would be in taking advantage of the time when the ship is end on to the battery : under these conditionxS we have greatly increased chances of hitting at 6S00 meters with the piece of IG cent, over those of hitting with the same piece at 3500 meters if the ship is parallel to the battery. If the fire is from one ship against another ship, there would be more chance ofhittiug the deck of the adversary in chase or in retreat: how= ever, we give him the same advantage : but if the two ships occupy the positions represented below, and they have the same number of gnns, =B C CD will have an enormous advantage over his ad= versary. in supposing that the armor cannot be penetrated by shots from a distance for which shell D fije is still verv efi'ectual. There is some interest in knowing the amount of the error we commit in neglecting, as we have done in the last case, a part of the length of the ship. If. remaining under the same conditions of fire, we determine the probability of hitting the parallelogram ahcd (Fig. 18), formed by the sides of the rectangle to be hit prolonged to the extreme errors, it will be greater than that which we have found by employing formula (23) by a quantity less than 0.01. A ship of the same width and more than 110 meters in length will be entirely comprised in this figure. This will evideutly not be the same if the ship is in the line of fire, but in this case we take account of the actual length of the ship. In other respects the use of the right line as the curve of errors introduces an error in the calculation ; but this correction, evaluated as it will be hereafter, gives, for all the cases which we have just examined, only inconsiderable quantities, of which it would be puerile to take account in practice. 46 -4— XYI. DATA FOR CALCULATION — APPLICATION TO THE SUPPLY OP MUNI- TIONS NECESSARY TO PRODUCE A GMYEN RESULT — PROBABILITY OF SUCCESS WHEN WE ONLY HAVE n PROJECTILES. We could continue indefinitely the applications of this method, their number is unlimited; but the examples already given are sufficient to enable us to comprehend it, and we should be able to calculate without difficulty the probability of hitting an object of any form with an ap- proximation very near to the value which we would obtain from the ordinary formula, supposing that it would enable us to solve the ques- tion. The certainty that we can easily find these values increases consider- ably the field of useful information that the calculus of probabilities is in a condition to furnish to practice; many questions which have been solved satisfactorily for other purposes become applicable to firing. There will be some modifications to be introduced on account of the nature of the problem. We will give one example to show the nature of these modifications. Often we know how many times it is necessary to strike an object with a given proj ectile in order to produce a certain result, which may be the ruin of an epaulement, the opening of a practicable breach, the destruction of a bomb-x)roof, the demoralization of a body of infantry or cavalry, &c. The experiences of the Island of Aix, of Graudentz, of Gavre, of Metz, <&:c., and those of former wars, furnish much informa- tion on this subject which it is easy to procure. Knowing, then, that it requires K projectiles which hit to produce the desired result, we can easily calculate how many it will be necessary to use in all, which is, as is evident, a question of the supply of munitions. The probability of hitting the object being j;, that of not hitting will be 1— jp or q; conse- quently if we fire n shots in succession, each of the terms of the develop- ment of (g'+i^)" will represent the probability of hitting zero times, once, twice, n times. Thus the probability of hitting at least once, at least twice, at least K times, will be unity less the first term, less the two first terms, less the K first terms ; in particular, that of hitting the object in n times will be that of hitting at least once, and we will have W p=i~r5 . and if we wish to know the number of projectiles which it is necessary to use in order to have a probability P of attaining the desired result, we must solve this equation for Uj which will give log(l-P) (c) w=,-''-), L ^ ' log l~i?) 47 The abbreviation log. means ordinary logarithms. If, for example, the fire is against one of the Moncrieff pits, of which we have spoken in XI Y, and under the conditions of the numerical example which it con- tains, conditions which assign to P the value 0.018, or simply O.Oo, we find the following results in making K=l ] that is to say, in admitting that a single shell, falling in the midst of the gun's crew crowded in the pit, and projecting its fragments upon a carriage as complicated as a disap- pearing one, would be sufficient to silence its fire. First. If one can use 30 shell, the probability of success will be 0.79, formula [h) ; 79 times out of 100 we will succeed in reducing the adver- sary to silence, or about four times out of five. Secondly. If we wish to know how many projectiles it wouid be neces- sary to fire in order to have the probability J of succeeding, we make P=J in the formula (c), and find ??=ll5 that is to say, if we only have 14 projectiles we will have equal chances of succeeding and failing. Thirdly. If we wish to consider the certainty of succeeding, we give P a large value, for example 0.92 ; the same formula shows that we would have to fire 50 shell under the conditions already given, in which we take ,-=3'^ and ^'=20'^. When K equals iico^ or when it will require two shell to attain the re- quired result, we have, by subtracting from unity the last two terms of the development of {p+g)" : {d) p=.l_(,i^g'-i + gn), and in the same manner for K equals 3, 4, &q. ; yet we can only deduce the value of n by a series of trials easily made. Applying formula (d) to the same case, if we wish to put two projectiles into a Moncrieff pit, we find the following results : With a number of projectiles equal to 100 50 30 we have the probabilities of success, / 0.96 0.63 0.45. r These methods are exact; but, when K is large the calculations are too difficult to be made directly. We can obviate this difficulty : the larger term of the development of (g+i^)'' represents evidently the prob- ability of the most probable case. K the exponent of 2? in this term is Z, the most probable case is when we hit the object I times out of n shots; 7 is, in other words, the greatest whole number contained in p (^? + l), and \fn were a little greater, we would have I =pn, nearly. The well- known formula, 2 / V2pqn can be considered as the sum of the terms (2?+^)^ of which the expo= nents of 2^ are comi)rised between l-^g and l-^-g. But if we make K 48 equal to l—g, the problem will be the determination from this formula of the probability of hitting the object a number of times comprised between l—g and n inclusive. The integral above, having its elements positive and negative equal two and two in absolute value, we can write -vVy « ■1/2 pqn 1/2 pqn We conclude that the probability of hitting the object a number of times comi)rised between I and l-~-g is equal to that of hitting it a num- ber of times comprised between I and l-\-gj and consequently the proba- bility of either is V'2 pqn But the probability of hitting the object at least K times is composed, first, of the probability of hitting it a number of times comprised be- tween I and l—g, and, second, a number of times comprised between I and w, which we can without inconvenience make equal to infinity 5 we can then write, "i/ 2 pqn -^/"--^/ «y Whence we write, ,., .... -i[i.^/?„].i±-^, in order that we may m ake use of the tabulated values of the integral which is contained in this formula, the calculation will then be extremely sim- ple. It is evident that ji)?*— K may be negative, and in that case we must subtract the numerical value of F (t) instead of adding it. This formula supposes that n is large, the degree of the approximation de- pends, then, upon the value ofp) if this quantity is very nearly zero it will give values of P too small, or it will require that n should be very large ; the contrary will take jjlace if ^ is nearly unity. This expression has a direct use since it enables an officer who has at his disposition n projectiles to see whether he is in a condition to obtain the desired re- sult ) we can use it otherwise for the solution of the problem which we proposed at first, that is to say, to find the number of projectiles which we must fire in order to hit the object at least K times with certainty 49 or with a probability, P. We must find in the table of F {t) the value of t which makes this integral equal to 2P— 1^ and, in placing, V2pqn ' we deduce n) but ordinarily we know the two values of n which corre- spond to P =i and P^l ; in the first case F {t) must be zero, and con- sequently from its upper limit equal to zero, we have (25) NJ=|; and if we have only »i projectiles at our disposition we will have as many chances of success as of failure. In the second case we must remark that P can only become equal to unity for ?i=co ; and, indeed, when there is the least chance of an event we cannot have mathematical cer- tainty of its accomplishment, but it is evident that we can make this adverse chance less than any given quantity by increasing the number of trials. Practically, we can admit that when we have eleven chances out of twelve of success the result should be considered as certain ; we choose this value of P, or, better, P=0. 92135, because it makes F {t) equal to 0.84270. We can then be assured that t is equal to unitj^ ; we have, then, (25), .... .^^^^g+K+V(g + Kf-i>-K-^ a formula which no longer requires the aid of a table; if ^ is sufficiently small, which often is the case in practice, this is reduced approximately to 2(g+K) (25)2 n^i= P Expressions (24) and (25), and (b) and (c) give the same information, and are applicable to all values of K ; but we should not forget that they are only approximations, and whenever it is possible we should make use of exact formulas. XYII. MODIFICATION OP THE PROBABILITY OP HITTINQ ON ACCOUNT OP CAUSES OP ERROR INDEPENDENT OP THE PIECE — NUMERICAL EX- AMPLE. When new causes of error independent of the piece come to be added to those for which have been calculated the mean error of the tables of fire, there results the usual principle of the theory of errors that the total mean error is the square root of the sum of the squares of the mean errors which each of these causes would jjroduce if it acted alone; if, 5782 4 50 for example, these causes act laterally, and they produce individually the mean errors 61, e^^ &c., the total mean error E will be if they act in the direction of the range, we will have in the same way Thus when we require the probability of hitting any object, by means of the preceding formulas, we should, in reality, put in place of y and y' the quantities E and E^ This supposes, however, that we can take for the curve of probability of these new errors that which is repre- sented by the formula of Laplace, and this will be the general case ; but this supposition is only formal; it is evident that the quantities e^ 62 . . ei' e^'^ &c., should be determined by experience. To fix the ideas we will take a numerical example. We fire from a battery with the tubed rifle of 24 cent, against an iron-clad of the same type as the Eichelieu, the projectile being the elongated shell of 120 Ml., with a charge of 2d> kil. The distance is 4,000 meters, for which the Memorial de VArtillerie de la Marine assigns a mean error in direc- tion of 1™.47, and in rangel9™.6 ; we have also the time of flight, 11^3. If there is no other cause of error, the probability of hitting this object when its axis is in the plane of fire will be 0.95, which we may call certainty ; but now we will take the actual conditions of the fight. In the first place, the distance is not known, and we must measure it. Suppose that the instrument employed gives, at 4,000 meters, a mean error of 150 meters, an amount that is not exaggerated if we consider that the measurement must be made as quickly as possible, the object being in motion. The ship is approaching or receding from the battery with a certain velocity, which we try to estimate in order to correct the sight ; but, in view of the few resources we have at our disposition to effect this object, a sensible error may be made. Admitting that the method employed permits a mean error of one knot, by which we assign to the ship a velocity which will be too great or too small by 0™.51 a second, and, as the shell is in the air 11^3, there will result a mean longitudinal error of 5°^.8, consequently the total mean error in range E' will be -2 V 19.6+150+5.8=151^.8. In the total mean error in direction it will be necessary to take account of the mean error committed in the correction for the wind. Suppose that the very coarse method which we are obliged to use in our batteries permits a mean error of 2™, we know, from the work of M. Helie upon this subject, when the wind is across the line of fire, the deviation d which it will produce can be represented by the following formula : <^=^(T-V^a)' 51 in which W is the velocity of the wind in meters ; T, the flight of the projectile in seconds } X, the range in meters j Y, the initial velocity in meters, which in this case is about 474™ ; a the angle of projection, equal in this actual case to 7° 44' 3C. In making W=2°^, we find cZ=4™.3, and this will be the mean lateral error which results from the imperfection of the method emplo^^ed of measuring the velocity of the wind ; if this is the only cause of error, we have Ei=/\/l.47+ 4.3=4^.5.* We have neglected the other causes of error heretofore, but now we will calculate the probability of hitting a rectangle of 98™ in length by 17™ in width, when the total mean errors are 151.8™ and 4.5™ ; and we find P= 0.18. Thus, in spite of these corrections, this probability, and consequently the useful effect, will be at least five times less than on a range where we can eliminate these causes of error ; it would require in general five times more projectiles, and time, to jiroduce the same result, and the piece only performs a small part of what it could be made to do. We can also verify the fact that the error in the distance alone reduces the probability to 0.20 j that caused by the error in the estimation of the velocity of the wind reduces it to 0.82. We see that a simple error in a correction which is sometimes entirely neglected produces an appre- ciable eflect, since it causes the loss of about ^ the useful effect. It is of interest to see what result we would arrive at if we neglect this cor- rection : the velocity of the wind being 10 meters a second, the formula of M. Helie shows that there will result from it a lateral deviation of 21™ J but this we would not consider a variable error, of which we can determine a mean to which the law of ordinary probability is applicable; it is evidently an accidental error, and it produces a deviation of 21™ to the right or left according to the direction of the wind. The mean errors do not change their values, consequently the probability sought *We find in the Lessons on Practical Mechanics, by General Morin, a table of the pressures of the wind upon an immovable plane surface of one square meter, result- ing from the experiments of the English physicist Rouse : for velocities of 8 and 10 meters the pressures are 7^.4 and ISi^..^, and their difference is GKl, consequently, if we make an error of two meters in the velocity of the wind in the case now before us, the error would be the same as one of G^^.l in a pressure on a surface of one square meter. Admitting, first, that upon a convex surface, like that of a projectile, the action of the wind would be the same as upon a plane surface equal to its section on the axis; and, second, that on account of the small velocity given to the projectile by the wind, its pressure will be constant, we can calculate approximately the deviation required : this will be the space j)assed over by a body whose mass is , and sub- jected to an effort of 6^.1 during the time t, which in this case is 11^3. This gives d = 4™. 52 is that of hitting a rectangle having its sides parallel to the direction of the errors, but whose center instead of being at the mean point of impact has for its co-ordinates 21™ and ; the rectangle to be hit will then be completely without that which contains all the shots, and the probability will be equal to zero. The ordinary formulas give a value expressed in decimals so small as not to have any practical value ; con- sequently this neglect is sufficient to reduce the probability from almost certainty, 0.95, to the certainty of failure, 0. There results from this a curious fact, that with a piece less precise, or if there exist other causes which increase the total mean lateral error, we would have more chances^ of hitting the object, since it would be in all, or in part, contained in the rectangle of extreme errors. Under certain circumstances these chances can have a sensible value ; thus, in the above case, supposing that the mean error in range remains tbe same, but that on account of some cause the total mean lateral error becomes 10 meters, we find that the probability of hitting the deck of the given ship, when we neglect taking into account the wind with a velocity of 10 meters, is 0.17 (8) YlII j that is, we hit about once in six times. We must guard against concluding from this, as is sometimes done, that in certain cases there is ,in advantage in the employment of pieces less perfect that those of mod- trn artillery ; this proves simply that it is indispensable to apply the correction for the velocity of the wind, for it is very evident that, coarse though the means employed may be, yet they enable us to estimate this velocity with sufficient exactness to give a probability far higher than 0.17, and that the advantage of precision in the piece becomes thus evi- dent. Thus, suppose that the mean error in this example, due to the evaluation of the velocity of the wind, is doubled, that is to say, that we make an error of 4™ in the velocity of the wind, more or less, an error we would hardly make if we remembered only two or three of the values given in the special tables for different forces of wind ; for instance, in the Aide- Memoire cfArtillerie de terre, we find that the probability would be less- ened by this error of 4™ to 0.54,* that is to say, we would have three times * We must here define the meaning of this mean error of 4™, which we have attrib- uted to the judgment of the observer. Suppose that a person having some ideas upon the velocity of the wind makes a very great number of estimations, and that, without his knowledge, for' otherwise his judgment would correct itself little by little, we note each time the mean velocity by an anemometer sufficiently correct; taking, then, all the trials where this velocity was lO'", or where it did not differ from this quantity by an amount sufficiently large to be considered, we would have the mean of the absolute errors committed. It is to this mean that we attribute the numerical value of 4™. Introducing it as a constant in the expression 7/ = ce— " < , we assume what is not perhaps completely justified. It would be necessary that the mean of the estimations should be about 10™, that is to say, that there should be an almost perfect equality between the sum of the positive and negative errors ; but this only experience can show. Not having any data for the question, we are obliged to em- I)loy this method, of which the exactness can be considered as probable, but not as certain ; nevertheless, we can evidently regard the result to which it leads as a rough approximation, and this will be sufficient for our imrpose. 53 more chances of hitting than in the preceding case, which reduces to nothing the paradoxical assertion with which we commenced. We should, moreover, remark that, in order to obtain the probability 0.17 without taking into account the wind, we must suppose an imaginary piece, like the gun of 24 cent., of which the mean error in direction is seven times greater, without the mean error in range being changed j but, in reality, this last would also have a value much greater, which would sensibly diminish the chances of hitting. We can easily see that with the shell-gun of 22 cent., rifled and hooped, which at 4,000 meters has a mean lateral error of about lO'", the chances of hitting will be almost nothing, if we do not take account of the wind. Hence if we tire with equal care two pieces of unequal precision, it is evident that the superiority will always be with the one whose mean errors are the smallest ; but the calculation gives the numerical value of this supe- rioritj' , and shows without difficulty that it is weaker as the corrections are less precise, so that, when the causes of error independent of the piece are very great in comparison with the mean errors, the chances of hitting become little different whichever piece we employ. We must not conceal from ourselves that the want of means of correction in which we are actually i3laces us nearly in this condition. When, instead of being in the i^lane of fire, the ship is j)erpendicular to it, we arrive at analogous results ; we remark that the velocity of the object will have in this case as marked an influence as that of the wind in the preceding discussion. We are struck with the enormous loss of useful effect in the fire of the pieces of the new artillery from a distance when we only have such means as we use at present for correcting them. The instrument is one of great precision, but the imperfect manner in which it is used renders this precision almost useless. The conclusion which we must draw from this is that we must reduce as much as possible the errors which we can commit in the instantaneous estimation of the distance, and also of the velocities of the object and of the wind. There is no question more urgent in requiring a solution than this, and we will admit without hesitation that, no matter how costly these means of estimation may be, at whatever price the instruments that would accomplish this purpose could be purchased and placed in our batteries, there would be a great economy, for nothing is more ruinous than the waste of our large modern projectiles. The figures at which we arrive in this paragraph show also the grave responsibility which rests on the officer who directs the fire j without doubt his talents and his experi- ence have a considerable influence upon the value of these corrections, which are of sufficient importance to change the certainty of hitting the object into the certainty of not hitting it. These reflections are particu arly applicable to fire at sea, since to the same causes of error are added that which we may make in our own speed ; they show, in both cases, what would be the true nature of a fight at a distance under these conditions. 54 XYIII. CONSIDERATION OF THE ERROR WHICH RESULTS FROM THE EM- PLOYMENT OF THE RiaHT LINE IN THE DETERMINATION OF THE PROBABILITY OF HAVING AN ERROR LESS THAN X, Before finishing this memoir we will make some useful developments of the nature of the error which is introduced in the calculation by the substitution of the right line for the ordinary curve of errors, as is shown in this figure. It is of interest to compare the ordinates of the Mg.m. curve and those of the right line, which, multiplied by dx^ represent the probability infinitely small of having exactly the error x. We will then compare the successive values of the probability of not exceeding the error x^ or the areas between the axis of s, that of x^ a variable ordinate, and either the curve ABO or the right line DE, the former being given by the ordinary formula, which, in terms of the mean error, can be put under the form : P= V- y^/TT e-i'dt=:f{t), and the latter, as we have seen (2), I, by p _2x x^ _2x x^ If we compare the successive values of P and Pi for the same values of -, the greatest difference is about 0.02, which corresponds to - equal to * In tlie note of Article III it was slioTrn that tlie mean error y was equal to 1 XT, „^„„„ 7 1 ^ 7„ ic therefore h- ll\/7t the usual form. rVr^^ and lix: y\/7t Making t=zhx, the above expression takes 55 1.35 and 2.70. The mean difference is 0.0125, which can be entirely neg- lected in practice, when we consider that the tables of fire from which we take the mean error have been compiled from a comparatively few rounds at each range, and that consequently the mean error is not de- termined with a degree of precision at all comparable with the above difference. If, however, we simply wish to determine the probability of not exceeding a given error, we would make use of the ordinary formula, for the value of which tables have been calculated j the em- ployment of the right line would in fact be useless if in practice we were only led to such simx)le questions. That we may know i)erfectly the nature of the approximation which we use, we will make a few more remarks which have some interest. If we call n the number of errors equal to CD, Fig. 2, and N the number of errors between zero and m, the ratio ^ will be equal to the ratio of the areas ydx and O AB ; the latter being equal to J, we have n=2^ydx*j the sum of the squares of the errors equal to x is nx"^^ or 2'Eo(^ydx) the sum of the squares of aU the errors comprised between zero and will be pin 51^ I x^ydx', we will obtain the mean of these squares by their number 'E) then the mean quadratic error will be the square root of the quotient 5 calling this quantity u^ and substituting for y its value, we have w= 2 / x'^^^^^dxx whence we find and w =2.4495 n. We have already m^'dy^ consequently, .2247. 56 We know that in using tlie/(^) we find that the ratio of the error of the mean to the mean error is V 1=1.2533* There is, as we see, a slight difference between the two values of this ratio. It results from this, that if the mean errors represent the same length in the/(^) and the formula we adopt, the square roots of the mean squares of the errors would not correspond exactly, and that to pass from one to the other it would be necessary, if we wish that the two formulas should really be functions of the same quantities, to adopt a mean value of — . It is evident that we could thus make equal the mean squares of the errors 5 but in this case the mean errors would not coincide; and that we would make use of a different right line than the one we have just employed, but, however, extremely near to it ; here would be but little advantage. We would find that the greatest differences would be equal to some thousandths. Although this diffi- culty is not serious, it may lead to some confusion. That is why, in *Tliis is obtained as follows. The value of the mean error, y, from the note to Article III, is y= = . If we call wi the value of u when 2/=^=e~^'*', we have V'^ Jo -^'^Vdo:. We can integrate this by parts 5 But •Joo -Job and ' 1 -• .-0 But 7^ 1/^ Jo Vtt Jo «■ -2A= and«.=^. Hence 1 C. A. S. 57 the precise calculatious the simi^ler way woiikl be to make use only of the mean error, it is sufficient foi- all tlie i)r('ceflin.i>- observations. Strangers appear to use almost exchisively the i)robable error; that is to say, tliat of which we have one chance out of two of not passing. This vahie we easily obtain by nmhing P equal to i in one of the formulas for the probability of not exceeding an error, x ; we find, by interpolation ^1=0.8453: * r with the right line we have from which we find and m 2m^ *^ Y ^=3(i_^ir) =0.8787. Y The difference 0.03 between the values of the ratio ^ occasions the r same remarks as the preceding. The advantage which results fi'om the employment of the probable error is the more simple value of the proba- bility of not passing it ; it is easily remembered, but it has not a value more definite than that of any other error, only the numerical values of these probabilities are not so easy to remember. If, in working to hundredths, we take the mean error as that which we have 57 chances out of 100 of not j)assing, we have tor practice information of the same value as that which the knowledge of the probable error furnishes; the latter will have, with the formulas of this Memoir, the inconvenience of introducing some numerical coefficients less simple, and as the rapidity and ease of calculation are of much importance, we have thought it bet- ter to continue to make use of the mean error. XIX. DETERMINATION OF THE CORRECTION TO BE APPLIED TO THE PROBA- BILITY OF HITTING ANY SURFACE IN A PRECISE CALCLTLATION. The formula which represents the proT^ability of hitting a rectangle having its center at the mean point of impact, and its sides parallel to the axes of the target, being the product of the two probabilities j)^ and p., which we have See note to Article III.— C. A. S. 58 of iu)t i>assing x aud 2/, we will have, calliDg z^ and £3 the errors of |)i and jjg, for the real value of the probability : the product £1 £3 can be neglected, since, after what we have vsaid, it can only represent some ten thousandths ; since P'= |>i^2 there remains for the appreciable part of the error attaching to P', A comparison of the values of P and P', in the preceding article shows that we obtain the greatest value of E when ^1 and 2)2 are each equal to 0.99 ; the ratio of the half sides of the rectangle to the mean errors will then be 2.7 and E will be very nearly equal to —0.04. When the figure to be hit is very near this rectangle, and when we consider that this error is too large, we can make the indicated correction afterward, but only in cases where the method emplo^^ed permits us to consider that we have arrived at a result sufficiently exact to warrant this correction. In the case of the rectangle we will not have this inconvenience, for tables have been prepared for the values of the/ if). General Didion in his CaleuUis of Prohahilities applied to fire^ 1858, gives one very complete, but the argument is the mean square of the errors, a quantity which we rarely have given. [Others have been calculated, however, with more convenient arguments, and one is given at theend of M. Breger^s Memoir, but it has not been reproduced in this translation.] ^M 3^0 >■> :M^ > 1^3> . 5 ' 3> ^^ ^^-^^>>'""^^>^^^ "^^»> :2>5^-^"-^ :>.":>>>> ■D"!);-. >3> 5->:^:^.2i^ - ^_!3>I>^' i5]yM >!>:$£> ^^i->^ 3)3 >l^ ^& >-M)'.,>X